Effective sample size

Summary

Effective sample size (ESS) estimates how many equally weighted samples a weighted particle population effectively contains:

$$\mathrm{ESS}=\frac{{\left({\sum }_i{w}_i\right)}^2}{{\sum }_i{w}_i^2}=\frac{1}{{\sum }_i{\tilde{w}}_i^2},$$

where $w_i$ are unnormalized weights and ${\tilde{w}}_i=w_i/{\sum }_j{w}_j$ are normalized weights. For $k$ particles, $1\le \mathrm{ESS}\le k$: values near $1$ indicate that a few particles dominate, whereas $k$ indicates uniform weights. Low ESS therefore diagnoses weight degeneracy and can trigger adaptive resampling (1). This is used in Feynman-Kac steering to decide when to resample.

1.

Singhal R, Horvitz Z, Teehan R, Ren M, Yu Z, McKeown K, et al. A General Framework for Inference-time Scaling and Steering of Diffusion Models. In: Proceedings of the 42nd International Conference on Machine Learning. PMLR; 2025. p. 55810–27. Available from: https://proceedings.mlr.press/v267/singhal25b.html